Exponential Families with Incompatible Statistics and Their Entropy Distance

This work generalizes the interaction measure of multi-information known in probability theory to finite-level quantum systems. This is done in the more general context of the entropy distance from an exponential family. One of the most well-known measures of stochastic dependence is multi-information, applied in various fields including Neuroscience and Statistical Mechanics. Multi-information is the entropy distance from an exponential family. In statistics, an exponential family is a very familiar parametric model, it admits a simple geometric description of entropy distance and maximum likelihood estimation by mean values. But a complete description of these concepts requires to studyThis work generalizes the interaction measure of multi-information known in probability theory to finite-level quantum systems. This is done in the more general context of the entropy distance from an exponential family. One of the most well-known measures of stochastic dependence is multi-information, applied in various fields including Neuroscience and Statistical Mechanics. Multi-information is the entropy distance from an exponential family. In statistics, an exponential family is a very familiar parametric model, it admits a simple geometric description of entropy distance and maximum likelihood estimation by mean values. But a complete description of these concepts requires to study extensions of a family, an investigation that was started by N. N. Cencov and O. Barndorff-Nielsen and continued principally by I. Csiszár and F. Matúš. Very little is known about extensions of an exponential family for a finite-level quantum system. In this thesis we consider mean value parameters of the statistic of such a family and a suitable extension thereof. Generalizing probability theory, we prove that the parameters describe the entropy distance from the family and they parametrize its rIclosure consisting of the points approximated in relative entropy. The dimension function of a local maximizer of entropy distance is bounded by the dimension of the family. We show that the rI-closure of a Gibbs family consists of the maximum entropy ensembles. A new and generic phenomenon of a non-abelian exponential family is the appearance of non-exposed faces of the convex mean value set. We prove that mean values of the closure of an e-geodesic included in the family meet only the relative interior of exposed faces. Unlike in finite probability spaces there are examples with a discontinuous entropy distance, the continuity being equivalent to equality of rI-closure and topological closure of the family. Examples suggest that the topology of an exponential family is related to the topology of associated projector lattices and to open projections and symmetrizations of state spaces. We conclude that multi-information for a quantum system is a continuous function equal to the entropy distance from a factorizable family. For analysis of a factorizable family we supply a partial classification of convex exponential families. As a perspective to a dynamical situation we examine a measure of temporal interaction for abelian systems, which is related to multi information.…